I have a comment concerning a proof of Menelaus's Theorem that is presented during the discussion of Einstein's remarks on elegant and ugly proofs (https://www.cut-the-knot.org/Generalization/MenelausByEinstein.shtml#Carnot). It is the proof that is marked "Proof #3" in the corresponding applet, and which serves as an impressive example of a proof which is elegant despite involving auxiliary constructions...

The proof starts "Draw a line perpendicular to the transversal EDF ...". My comment is this: the proof does not seem to make, actually, any use of the fact that the auxiliary line (call it l₁) is perpendicular to the transversal. The proof does use the following principle: "the segments cut on two lines by a family of parallel lines are in the same ratio". Therefore it is required that the segments aKa, bKb, cKc are all parallel to the transversal. But for that they don't seem to have to be perpendicular to l₁. The only requirement from l₁ seems to be that it intersects the transversal EDF.

Or perhaps I've missed something.

Ram

No, that's a good remark. The only thing that matters is that the lines through the vertices come out parallel to the transversal. Surely this might be the starting point with a line crossing the four as the second stage.

Many thanks,
Alex